Velocity estimation technique using seismic data is often based on time/distance equations which are independent of direction, and even though we now know that many rocks are quite anisotropic, useful results have been obtained over the years from these isotropic estimates. Nevertheless, if velocities are significantly direction-dependent, then the isotropic assumption may lead to serious structural interpretation errors. Additionally, information on angle-dependence may lead to a better understanding of the lithology of the rocks under measurement. VSP and cross-well data may each lack the necessary aperture to estimate more than two velocity parameters for each wave type, and if the data straddle a symmetry axis, then these may be usefully chosen to be the direct velocities (from time-and-distance measurements along the axis) and NMO velocities (from differential time-offset measurements). These sets of two parameters define ellipses, and provide intermediate models for the variation of velocity with angle which can later be assembled and translated into estimates of the elastic moduli of the rocks under scrutiny.If the aperture of the measurements is large enough e.g. we have access to both VSP and cross-well data, we divide the procedure into two independent steps, first fitting best ellipses around one symmetry axis and then fitting another set around the orthogonal axis. These sets of four elliptical parameters are then combined into a new, double elliptical approximation. This approximation keeps the useful properties of elliptical anisotropy, in particular the simple relation between group and phase velocities which simplifies the route from the traveltimes measurements to the elastic constants of the medium.The inversion proposed in this paper is a simple extension of well-known isotropic schemes and it is conceptually identical for all wave types. Examples are shown to illustrate the application of the technique to cross-well synthetic and field P-wave data. The examples demonstrate three important points: (a) When velocity anisotropy is estimated by iterative Paper presented in two parts at the 53rd EAEG meeting, Florence, techniques such as conjugate gradients, early termination of the iterations may produce artificial anisotropy. (b) Different components of the velocity are subject to different type of artifacts because of differences in ray coverage. (c) Even though most rocks do not have elliptical dispersion relations, our elliptical schemes represent a useful intermediate step in the full characterization of the elastic properties.

We combine various methods to estimate fracture orientation in a carbonate reservoir located in southwest Venezuela. The methods we apply include the 2-D rotation analysis of 2-D P-S data along three different azimuths, amplitude‐variation‐with‐offset (AVO) of 2-D P-wave data along the same three azimuths, normal‐moveout (NMO) analysis of the same 2-D data, and both 3-D azimuthal AVO and NMO analysis of 3-D P-wave data recorded in the same field. The results of all methods are compared against measures of fracture orientation obtained from Formation microScanner logs recorded at four different locations in the field, regional and local measures of maximum horizontal stress, and the alignment of the major faults that cross the field. P-S data yield fracture orientations that follow the regional trend of the maximum horizontal stress, and are consistent with fracture orientations measured in the wells around the carbonate reservoir. Azimuthal AVO analysis yields a similar regional trend as that obtained from the P-S data, but the resolution is lower. Local variations in fracture orientation derived from 3-D AVO show good correlation with local structural changes. In contrast, due to the influence of a variety of factors, including azimuthal anisotropy and lateral heterogeneity in the overburden, azimuthal NMO analysis over the 3-D P-wave data yields different orientations compared to those obtained by other methods. It is too early to say which particular method is more appropriate and reliable for fracture characterization. The answer will depend on factors that range from local geological conditions to additional costs for acquiring new information.

I perform singular value decomposition (SVD) on the matrices that result in tomographic velocity estimation from cross‐well traveltimes in isotropic and anisotropic media. The slowness model is parameterized in four ways: One‐dimensional (1-D) isotropic, 1-D anisotropic, two‐dimensional (2-D) isotropic, and 2-D anisotropic. The singular value distribution is different for the different parameterizations. One‐dimensional isotropic models can be resolved well but the resolution of the data is poor. One‐dimensional anisotropic models can also be resolved well except for some variations in the vertical component of the slowness that are not sensitive to the data. In 2-D isotropic models, “pure” lateral variations are not sensitive to the data, and when anisotropy is introduced, the result is that the horizontal and vertical component of the slowness cannot be estimated with the same spatial resolution because the null space is mostly related to horizontal and high frequency variations in the vertical component of the slowness. Since the distribution of singular values varies depending on the parametrization used, the effect of conventional regularization procedures in the final solution may also vary. When the model is isotropic, regularization translates into smoothness, and when the model is anisotropic regularization not only smooths but may also alter the anisotropy in the solution.

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The problem consists of determining the unknown coefficients an from the measured traveltimes. Once these coefficients have been calculated, the computation of the sum (1) is straightforward.The representation (1) has two important degrees of freedom that influence decisively the kind of results obtained. These are the number (M) and kind of functions I3n(r) to be used. The most common choice for the functions f3n(r) is orthogonal cells (square or cubic pixels), and in that case the coefficients an represent the slowness within each cell (MeMeehan, 1983;Ivansson, 1985). Although this is the most popular basis function for estimating the slowness model, others have been suggested recently. Harlan (1989) defines the velocity function as a sum of smooth basis functions (Gaussians), and Van Trier (1988) defines the functions f3n(r) as cubic B-splines multiplied by functions that reproduce the expected structure of the model. The number of functions M is also arbitrary but is usually small to avoid having to solve a huge system of equations.The kind and number of functions used for expanding the slowness model determine many of the general features of the final image. With the same data set it is possible to obtain different results just because different parameterizations have been used. However, the goal is to obtain a reconstructed model free from these artifacts of parameterization, which means the selection of the basis function is critical in the inversion process and thus should be considered more carefully, as described below.There are no general criteria for deciding which representation is the best, although some may have clear advantages for solving specific problems. Our selection of the basis function will be based on minimization of the expression that estimates the norm of the null space of the problem ABSTRACT Traditionally in the problem of tomographic traveltime inversion, the model is divided into a number of rectangular cells of constant slowness. Inversion consists of finding these constant values using the measured traveltimes. The inversion process can demand a large computational effort if a high-resolution result is desired.We show how to use a different kind of parameterization of the model based on beam propagation paths. This parameterization is obtained within the framework of reconstruction in Hilbert spaces by minimizing the error between the true model and the estimated model. The traveltimes are interpreted as the projections of the slowness along the beampaths. Although the actual beampaths are described by complicated spatial functions, we simplify the computations by approximating these functions with functions of constant width and height, i.e., "fat" rays, which collectively form a basis set of natural pixels.With a simple numerical example we demonstrate that the main advantage of this parameterization. compared with the traditional decomposition of the model in rectangular pixels, is that 2-D reconstructed images of similar quality can be obtained with considerably less computat...